Wednesday, December 23, 2015

Often when we are watching science fiction shows or movies, they imagine fantastic devices that are just at the edge of possibility according to modern physics. Ideas like death stars, faster-than-light travel, or teleportation are all things we are unlikely to see for many years, but can’t totally dismiss as potential inventions of the far future. But every once in a while, science fiction posits something so totally absurd that I can’t help but laugh. The other day, while watching an episode of Star Trek: Deep Space Nine, I saw an idea in this category: a small device that would alter the laws of probability.

The plot of this episode, titled “Rivals”, involved a series of strange events on a space station. Gamblers would win against impossible odds, the infirmary suddenly filled with victims of freak accidents, a computer search of an unsorted file list would instantly discover just the right data, and a crew member who stunk at racquetball suddenly started making impossible trick shots. As expected for a Star Trek episode, this all boiled down to an alien technological device, in this case one that could alter the laws of probability. Once the captain found and destroyed the device, everything could go back to normal.

Now at first you might just label this as another piece of random technobabble used to advance a sci-fi plot. But I think this hits at a popular misconception about probability. Many people think then when the odds of some event are low, it’s some kind of conspiracy of the universe against them. So if you have a one-in-a-million chance of making a trick racquetball shot, why shouldn’t some gizmo be able to alter that probability and help your game? But actually, the low probability just reflects the fact that there are a million different ways you can hit the ball, all of which are equally valid executions of the laws of physics. The universe doesn’t really care about the one shot that we abstractly label the great trick shot: it’s just another in a huge sea of possibilities. Tiny variations in the angle of your aim and the force of your swing can make a major difference in where the ball goes. You can imagine a million parallel universes in which you hit the ball, and only one of them involves you making the shot successfully. Which one is it? You might care, but it’s none of the universe’s business. All are roughly equally likely, depending on your exact position and momentum when you hit the ball.

So what would it mean for an alien device to change the laws of probability? It is theoretically possible for some specific technical gizmo attached to your arm to bias it towards the successful racquetball shot. But a generalized probability gizmo that could enhance your shot, improve computer data searches, enable victories at roulette, and cause freak accidents? How would this device know how to bias the universe precisely in ways that we label as “unlikely” results, in all these diverse domains? In the racquetball shot example, we’re estimating that there are a million possibilities, so *every* shot you take will be a one-in-a-million result: no matter what ends up happening, there was only that same tiny chance of that specific shot occurring. So a machine that caused an “unlikely” result for events would be useless at choosing the victory shot for you— every non-victory shot is equally as likely, and there are 999,999 of them. On the outer fringe of possibility, perhaps if the machine had full artificial intelligence capability, there might be possibilities here. But then it would be a techno-gremlin notable mainly for its intentional meddling in other’s lives, and nobody would describe this strange robot’s actions as a change in the laws of probability.

We can see similar laws at work in many parts of our daily lives. Is my daughter engaged in a specific effort to make her room messy? If I don’t enter my daughter’s room for a few weeks, books, toys, and clothes are strewn about everywhere, with barely a path available to the bed. But is she deliberately trying to make the room messy? Though there is some doubt about her intentions here, I think it’s really a result of the fact that there are many more configurations of the room that are messy than non-messy. You can put a sock in 1000 locations that are not the sock drawer, but need to spend some energy to intentionally put it in the sock drawer if that’s what you want. Combine all the objects in the room, and it seems there are uncountably more ways for the room to be messy than clean. Thus, without intentional action to drive it towards one of these clean configurations, small continuous changes will probabilistically lead to complete messiness. So she doesn’t need to be trying, the messy room is just something that will happen with high probability unless someone invests energy to prevent it. This is similar to the basic principles that the thermodynamic laws of entropy are built upon, though I won’t say much more about that right now, due to the large number of websites that seem to admonish us against the messy-room metaphor for this concept.

So, in short, when some event we want has a low probability of occurring, it is usually just a measure of the fact that there are many possibilities of what can occur, and only a small number have property that is significant to our human interpretations. The universe doesn’t know which one we want, so it has no particular reason to deliver the desired outcome. Imagining a technical device that can alter the laws of probability is like imagining a device that can make 2+2=5, or can cause triangles in a Euclidean plane to have angles totaling 190 degrees: it simply violates the fundamental mathematics of the situation. Over the next few millennia, humanity may see miraculous inventions such as laser pistols, teleportation, starships, or halfway decent William Shatner albums, but I think we can safely bet that no machine will ever alter the laws of probability. If you’re frustrated sometimes by things not going your way, you should take comfort in the fact that there are many ways things can go, and figure out what you can do to reduce the number of possible bad outcomes. The universe is not engaged in some kind of conspiracy against you that needs to be fixed with a magical gadget.

Sunday, November 29, 2015

With the holiday season upon us, many of you out there are probably giving or getting new tech gadgets as gifts. Once you unwrap your fancy new iPad, iPhone, or Android tablet, you’re probably asking yourself, “Now what do I do with this?” While you will probably be downloading lots of fun games and apps, you will somehow need to justify all the hours you spend in front of the screen to your family. One of the best ways to do this is to install a few math-related games, and provide some educational value for your children. But there is a bewildering array of supposedly educational games available for these systems. How do you know which ones to get? Today I will share some suggestions based on my experiences.

The first thing I should point out is that there are hundreds of games out there that are basically glamorized flashcards, presenting math problems directly and giving some kind of in-game reward for correct solutions. For example, they will put up a math problem, like “What is 5 x 5?”, and if the answer is correct, the player gets a few points. These points can then be traded in for virtual stickers, virtual ammunition against alien robots, or similar rewards. While there is nothing wrong with this type of game, and they have the advantage of being able to easily draw on large libraries of problems for different skill levels, I don’t find them very exciting. My daughter will play them if I tell her she needs to practice her math, but doesn’t usually come to me asking to play them. What I really want are original games that both teach math and can stand on their own as fun games. Fortunately, I have found two games that fit these requirements.

The first game I want to highlight is called “DragonBox Elements”. This game is designed to teach the basics of geometric proofs, a seemingly advanced topic, but they present it in a very accessible and intuitive way. Each basic shape, triangles and quadrilaterals, can summon a basic type of monster related to that shape. So if you can identify a quadrilateral among the shapes on the screen, you trace it out and summon a quadrilateral-monster. Line segments and angles are also marked with colors, such that any two objects with the same color have equal length, and you can upgrade the monsters to “special” ones using these. So, for example, if you notice a triangle-monster has two equal sides, you can click on them to upgrade to the slightly more powerful isoceles-monster. The monsters also have powers, which essentially invert this process: so if you have been given an isosceles-monster and its two equal sides are not yet colored, you can click on the monster and the two sides to mark them as equal. They also introduce other powers related to ideas like opposite angles, radii of circles, and parallel lines. So the basic Euclidean concepts of definitions, axioms, and theorems have been transformed into monsters and powers. I’m not totally sure how this will translate to actual proof skills when my daughter reaches that level of math class, but laying the foundations at such a young age can’t hurt. And more importantly, she loves this game, even asking to replay all the levels at “hard” difficulty after beating it once.

The second truly engaging iPad math game I have discovered is called “Calculords”. This is a card game, where each turn you have a bunch of cards in hand that you can use to summon creatures for battle. There are two types of cards, number cards and creature cards. It’s not a simple energy system like in most popular collectible card games though: in order to summon a creature for battle, you need to add, subtract, and multiply number cards to reach the creature’s number. The creatures are then placed on a lane-based battlefield, where they fight the evil monsters summoned by an alien enemy. For example, suppose you have a Hungry Blob card, a monster with a summoning cost of 15, and your number cards are 3, 3, 4, and 1. You can form a 15 using 3 x 4 + 3, so you can play those cards to summon your blob. But an additional wrinkle, adding to the mathematical challenge, is that you also gain extra bonuses if you precisely use up all your number or creature cards. So a better move would be to play 3 x 4 x 1 + 3, which still reaches your 15, but uses up your numbers. Since you have 9 creature cards and 9 number cards on each turn, the number of potential choices and calculations is quite large, and the strategy to summon the best set of monsters while trying to use up cards to get the bonus can get very involved. But the game offers many enemies at a variety of difficulty levels; my daughter has been playing at the easier levels since she was in 2nd grade. This is another game that she and I have found quite addictive, and an amazing way to get her to eagerly practice her basic arithmetic. And at the top difficulty levels, even I find it challenging, when I sneak in a chance to play on my own.

So, in short, these are the two truly original smartphone/tablet math games I currently recommend for elementary-age students: DragonBox Elements and Calculords. Naturally, these are heavily influenced by my 4th-grade daughter’s tastes, and their effectiveness probably varies a lot at older and younger ages. DragonBox elements provides the amusing and engaging transformation of Euclidean definitions, axioms, and theorems into monsters and powers. And Calculords provides a strategic challenge involving arithmetic calculations that is accessible to young children at lower levels, and fun even for adult math geeks at the hardest settings. If you have kids at the upper elementary level who could use some extra math practice, be sure to take a look at these excellent games. Also be sure to post reviews on iTunes or similar sites if you like them, as this will increase the chance of further games appearing from these talented authors.. And as always, I’ll be interested to hear from you on this topic: with such an overwhelming number of smartphone and tablet games out there, I’m sure there are a few great ones that I haven’t discovered yet.

Sunday, October 25, 2015

Rudolf Steiner was a prolific Austrian author and philosopher of the late 19th and early 20th centuries. He felt a strong connection to mysticism and spiritualism, ever since he supposedly communicated with the ghost of a recently deceased aunt at the age of 9. Steiner is well-known for having led a group that split off from the popular circle of European mystics known as the Theosophical Society, which seemed heavily inclined to regard the religions of East Asia as somehow providing the keys to understanding spirituality. Steiner called his new group the Anthroposophical Society, and this competing group believed that Western science and culture were just as strongly connected to the spiritual-- it was just a matter of intepreting them properly. One particular Western idea that Steiner was fond of was the concept of a fourth physical dimension, another mathematically-defined direction that we cannot percieve but is just as real as length, width, and height. Steiner believed that our consciousness extended into this fourth dimension, and that phenomena like ghosts and ESP resulted from activity in this hidden dimension. And most interestingly, he believed he had a simple philosophical proof that this fourth dimension really does exist, and our human minds really do extend into this additional dimension.

Here's how Steiner's proof goes. We all know that a creature of a particular dimension, if it looks out at its world, really only sees a view that is one dimension smaller. For example, a one-dimensional creature living in Lineland, a universe that exists entirely on a single straight line, can only perceive a single point on either side of himself: a zero-dimensional view. Similarly, a two-dimensional Flatlander, living on a plane, really only sees a line; it is only us three-dimensional creatures, looking down on the plane from above, who can truly comprehend its full two-dimensional world. And in real life, when we look out with our eyes, we are only seeing a plane. Yet somehow we do believe we fully understand and perceive the three dimensions of our world. Steiner draws what he believes is a natural conclusion from this: "The fact that we can delineate external beings in three dimensions and manipulate three-dimensional spaces means that we ourselves must be four-dimensional... We float in a sea of the fourth dimension just like ice cubes on water." In other words, our ability to fully perceive our three-dimensional space shows that our minds must extend beyond those three dimensions.

It's a fun thought, but you can see something fishy there right away, if you think about the world of modern computing. I can think of all sorts of situations in which an object in three dimensions is represented by a model in fewer dimensions. For example, most computer memories and circuits that power modern three-dimensional computer games are essentially stored in flat two-dimensional circuit boards. While these are technically 3-D like all physical objects, the memory storage can be thought of as truly two-dimensional in some sense, as each (x,y) coordinate on the circuit board only stores one encoded value at any given time. More basically, you may recall the concept of a Turing Machine discussed in some earlier podcasts: this is a theoretical model of computing, based on writing and reading values from a long, essentially one-dimensional, tape. It has been shown that any modern computer can be modelled by a very slow, but 100% accurate, Turing machine equivalent. So even the 3-D models in a modern computer game could, with enough work, be represented in one dimension.

I think the main flaw in Steiner's argument is his fundamental premise, that a creature of n dimensions can only perceive n-1 dimensions. It is true that through the sense of sight, a creature can only see one dimension lower, but our senses are not limited to sight. Think about a blind man, who perceives the world mainly by walking around and tapping items with his cane to understand their form: he can walk forward, back, right, or left, and even climb ladders up and down. He is truly perceiving the full three dimensions of his world, travelling within all three of those dimensions and building a mental model based on his real experiences. This applies to the lower-dimensional examples as well: the flatlander can move around and perceive his full plane, and even the poor Linelander can move back and forth on his line. Thus, the idea that perceiving your full dimensionality requires capabilities from a greater dimensionality does not really seem to ring true. You need to think of perception much more generally than simple line-of-sight.

Naturally, this does not fundamentally prove that Steiner was wrong about our minds extending into the fourth dimension; it just means that the proof of such an idea is not so simple. So it's still entirely possible that the concept of our mystical four-dimensional minds is correct but unproven, and the rest of Steiner's Anthroposophical Society ideas might still be valid. This philosophy of the fourth dimension was just a launching point for a variety of mystic concepts, related to traveling along this fourth dimension to the astral plane where you could encounter ghosts, life after death, etc. Some of Steiner's lectures get amusingly specific on details of the astral plane-- apparently he believed that his meditation and similar activities had actually taken him to this place, so he could talk about how astral dimensions mirrored our own, and writing there would appear backwards. Personally, I'm a bit of a skeptic on this topic, but these kinds of ideas do seem to have a lasting appeal, as shown by the New Age sections you can find in many modern bookstores. If you're into that stuff, try meditating hard enough, and maybe you too can follow Steiner's path into the astral plane through the fourth dimension. While you're there. see if you can track down Steiner's spirit, to discuss the flaws in his philosophical proofs.

Sunday, August 30, 2015

Those of you who follow the Math Mutation facebook feed may have noticed that a book I co-authored was just released: "Formal Verification: An Essential Toolkit for Modern VLSI Design". Now, I need to caution you that this is not a Math Mutation book-- it's a technical book, aimed at electrical and computer engineers invovled in chip design. However, I do think Math Mutation listeners might have some interest in the core concepts on which the book is based. So, today I'll try to give you a brief summary of what Formal Verification is, and why it's worth writing a book about.

You're probably aware that modern computer chips are pretty complicated, by many measures the most complex devices ever created by man. For new ones coming out this year, the number of transistors is measured in the billions. So it makes sense to ask the question: how do we know these things will work? It would be prohibitively expensive to just build them first and then test them, so we need to discover and fix as many of the bugs as possible during the design stage. The process of finding these bugs is known as design validation. For many years, the most popular technology used in design validation has been simulation: testing how a software model of the design will behave for various sets of inputs.

Unfortunately, even a simple chip design has so many possible behaviors that there is no way to simulate them all. For example, suppose you are designing a simple integer adder: it will take 2 numbers as inputs, each expressed with 32 bits, or binary digits, which can each be 1 or 0. It will then output their sum. How many possible input sets are there for this design? Since each input has 32 bits, it has 2^32 possible values, thus resulting in 2^64 possible values for the pair. If we assume we have a fast simulator that can check 2^20 values per second, that means that we will need 2^44 seconds to check all possible values-- over half a million years. Most managers are not very happy when given a time estimate on this order to finish a project. And needless to say, most chips sold today are many orders of magnitude more complex than a simple adder.

So, what can we do to make sure our chip designs really will work? A variety of technologies have been developed over the past few decades to try to find a good set of example values to simulate. And they do seem to be doing a decent job: most electronic devices you buy today seem to more-or-less do what you want them to. But it still seems like there should be a better way to validate them: no matter how good you make it, simulation and related methods can never cover more than a tiny portion of your design's possible behaviors.

That's where formal verification comes in. The idea of formal verification is to take a totally different approach: instead of trying specific values for your design, why not just mathematically prove that it will always be correct? That way you don't have to worry about trying every possible test case. If there is a single set of values that would generate an incorrect result, your proof will fail, and you will know your design has a bug. If you do succeed in mathematically proving your design correct, then you know that there is no bug, and do not need to waste time simulating lots of testcases. In effect, formally verifying a design is equivalent to simulating all possible values. Many would argue that philosophically, this is really the "right" way to validate chip designs. You may have heard the famous Guindon quote, "Writing is nature’s way of letting you know how sloppy your thinking is." Formal Verification pioneer Leslie Lamport expanded on this with "Math is nature's way of letting you know how sloppy your writing is.", and later added "Formal math is nature's way of letting you know how sloppy your math is."

You've probably guessed by now that there has to be a catch. Formal verification is easier defined than done: when billions of transistors are involved, how do we even get our heads around the problem of creating mathematical proofs? It's far beyond what anyone could manually do, so to make this method a possibility, humans need to be aided by intelligent software that helps to automate proofs. To further complicate matters, it's also been shown that any formal verification system needs to internally solve what are known as NP-complete problems. If you remember our discussion way back in episode 13, an NP-complete problem is "provably hard" in some sense, meaning that no piece of computer software can ever solve it efficiently in 100% of cases. However, researchers have worked for many years to try to develop practical software that could utilize clever tricks to enable real proofs on a wide variety of actual industrial product designs.

The good news is that, in the past decade, formal technology has advanced to the point where it really is practical for an average design engineer to use in many cases. While formal verification software can't handle full multi-billion-transistor chip designs, it can often enable an engineer to create solid proofs on major sub-blocks that go into a chip design, massively reducing overall risk of bugs. Using formal verification software remains a bit of an art though. Due to the NP-completeness issue, the software may get stuck or progress very slowly: the user must often give subtle hints and suggest shortcuts to enable the proofs to complete. In addition, formal verification is a problem that is impossible to fully automate: no matter how good your software gets at proving stuff, a human still has to somehow be able to tell it what stuff to prove-- what is the overall intent of the design in the first place? Ultimately, someone has to carefully transfer the design intent from their human brain into a machine-readable form, and understand the possible limitations and pitfalls in this process. Sadly, computer software that directly plugs into your brain is probably still many years away, and even then I have the feeling that many of us think too sloppily to enable this level of verification directly.

Thus, the need for a Formal Verification book. While there have been many books on Formal Verification published over the past few decades, most have focused on internal algorithms that would be needed to develop the software involved. Our book is one of the first real practical manuals designed to help deesign and validation engineers use formal verification software on real-life design targets. Anyway, that quick summary should give you an idea of what our new book is about. If you're in a field where you do chip design or something related, please visit our book's website at http://formalverificationbook.com, order a copy, and tell all your friends about it! If you're not in this field, the book probably won't make much sense to you, but hopefully you've still enjoyed this episode of the podcast.

Sunday, January 18, 2015

Before we start, I'd like to thank listeners katenmkate and EdB, who recently posted nice reviews on iTunes. I'd also like to welcome our many new listeners-- from the hits on the Facebook page, I'm guessing a bunch of you out there just got new smartphones for Xmas and started listening to podcasts. Remember, posting good reviews on iTunes helps spread the word about Math Mutation, as well as motivating me to get to work on the next episode.

Anyway, on to today's topic. We often think of mathematical history as something that happened far in the past, rather than something that is still going on. This is understandable to some degree, as until you get to the most advanced level of college math classes, you generally are learning about discoveries and theorems proven centuries ago. But even since this podcast began in 2007, the mathematical world has not stood still. In particular, way back in episode 12, we discussed the strange case of Grigori Perelman, the Russian genius who had refused the Fields Medal, widely viewed as math's equivalent of the Nobel Prize. Perelman is still alive, and his saga has just continued to get more bizarre.

As you may recall, Grigori Perelman was the first person to solve one of the Clay Institute's celebrated "Millennium Problems", a set of major problems identified by leading mathematicians in the year 2000 as key challenges for the 21st century. Just two years later, Perelman posted a series of internet articles containing a proof of the Poincare Conjecture, a millennium problem involving the shapes of certain multidimensional spaces. But because he had posted it on the internet instead of in a refereed journal, there was some confusion about when or how he would qualify for the prize. And amid this controversy, a group of Chinese mathematicians published a journal article claiming they had completed the proof, apparently claiming credit for themselves for solving this problem. The confusion was compounded by the fact that so few mathematicians in the world could fully understand the proof to begin with. Apparently all this bickering left a bitter taste in Perelman's mouth, and even though he was selected to receive the Fields Medal, he refused it, quit professional mathematics altogether, and moved back to Russia to quietly live with his mother.

That was pretty much where things stood at the time we discussed Perelman in podcast 12. My curiosity about his fate was revived a few months ago when I read Masha Gessen's excellent biography of Perlman, "Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century". It gives a great overview of Perelman's early life, where he became a superstar in Russian math competitions but still had to contend with Soviet anti-semitism when moving on to university level. It also continues a little beyond the events of 2006, describing a somewhat happy postscript: eventually the competing group of Chinese mathematicians retitled their paper " Hamilton–Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture", explicitly removing any attempt to claim credit for the proof, and recasting their contribution as merely providing a more readable explanation of Perelman's proof. Sadly, this did not cause Perelman to rejoin the mathematical community: he has continued to live in poverty and seclusion with his mother, remaining retired from mathematics and refusing any kind of interviews with the media.

As you would expect, this reclusiveness just served to pique the curiosity of the world media, and there were many attempts to get him to give interviews or return to public life. Even when researching her biography, Masha Gessen was unable to get an interview. In 2010, the Clay institute finally decided to officially award him the million dollar prize for solving the Poincare Conjecture There had been some concern that his refusal to publish in a traditional journal would disqualify him for the prize, but the Institute seemed willing to modify the rules in this case. Still, Perelman refused to accept the prize or rejoin the mathematical community. He claimed that this was partially because he thought Richard Hamilton, another mathematician whose work he had built upon for the proof, was just as deserving as he was. He also said that "the main reason is my disagreement with the organized mathematical community. I don't like their decisions, I consider them unjust." Responding to a persistent reporter through the closed door of his apartment, he later clarified that he didn't want "to be on display like an animal in a zoo." Even more paradoxically, he added "I'm not a hero of mathematics. I'm not even that successful." Perhaps he just holds himself and everyone else to impossibly high standards.

Meanwhile, Perelman's elusiveness to the media has continued. In 2011 a Russian studio filmed a documentary about him, again without cooperation or participation from Perelman himself. A Russian journalist named Alexander Zabrovsky claimed later that year to have successfully interviewed Perelman and published a report, but experienced analysts, including biographer Masha Gessen, poked that report full of holes, pointing out various unlikely statements and contradictions. One critic provided the amusing summary "All those thoughts about nanotechnologies and the ideas of filling hollowness look like rabbi's thoughts about pork flavor properties." A more believable 2012 article by journalist Brett Forrest describes a brief, and rather unenlightening, conversation he was able to have with Perelman after staking out his apartment for several days and finally catching him while the mathematician and his mother were out for a walk.

Probably the most intriguing possibility here is that Perelman has not actually abandoned mathematics, but has merely abandoned the organized research community, and is using his seclusion to quietly work on the problems that truly interest him. Fellow mathematician Yakov Eliashberg claimed in 2007 that Perelman had privately confided that he was working on some new problems, but did not yet have any results worth reporting. Meanwhile, Perelman continues to ignore the world around him, as he and his mother quietly live in their small apartment in St Petersburg, Russia. Something tells me that this not quite the end of the Perelman story, or of his contributions to mathematics.

Note that this podcast is intended mainly for audio consumption, so you will not see the numerous illustrations & diagrams you would find at most math sites, though these are linked in the show notes whenever possible.

The Math Mutation Book

The Math Mutation book, "Math Mutation Classics: Exploring Interesting, Fun, and Weird Corners of Mathematics", is now available. You can order it from Amazon at this link.

By the way, if you like it, don't forget to post a review on Amazon. And of course, if you're in the Portland, Oregon metro area, I'll be happy to autograph your copy sometime.

Donations

If you enjoy this podcast enough to donate money, thanks! But since I don't spend an appreciable amount on creating it, I ask instead that you donate to your favorite charity, and send me an email about it. You will then experience fame and fortune as I mention your name on the next podcast.